When do modules mimic arbitrary sets? (Q6600733)

The author study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism \(A \to B\), there is a monomorphism \(B \to A\) (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, it is proved that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky's dual rings) is right Artinian and right serial. Consequently, \(R\) is an Artinian PIR if and only if every factor ring of \(R\) cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. It is proved that a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other.